Cancellation of anomalies in a path integral formulation for classical field theories D. Mauro Department of Theoretical Physics, University of Trieste Strada Costiera 11, Miramare-Grignano, 34014 Trieste, Italy Some symmetries can be broken in the quantization process (anomalies) and this breaking is signalled by a non-invariance of the quantum path integral measure. In this talk we show that it is possible to formulate also classical field theories via path integral techniques. The associated classical functional measure is larger than the quantum one, because it includes some auxiliary fields. For a fermion coupled with a gauge field we prove that the way these auxiliary fields transform compensates exactly the Jacobian which arises from the transformation of the fields appearing in the quantum measure. This cancels the quantum anomaly and restores the symmetry at the classical level. PACS: 11.30.Rd, 45.20 Key words: chiral anomalies, classical and quantum functional methods 1 Introduction It is well-known that in field theory some symmetries could be broken in the quantization process. This is the phenomenon of anomalies and it is signalled by the non-invariance of the quantum functional measure under the symmetry transformations. In Ref. [1] it is shown that it is possible to formulate also classical mechanics via path integrals. We will indicate this formalism with CPI, for Classical Path Integral. This formulation has been extended to bosonic field theories in [2]. In this talk we will review Ref. [3] where this approach has been extended also to a theory of fermions coupled with a gauge field which, at the quantum level, present a chiral anomaly. As anomalies are not present at the classical level there must be a mechanism that cancels the anomaly in the CPI. In Ref. [3] this mechanism has been linked with the presence of several auxiliary fields in the CPI. These fields generate a Jacobian which compensates the one arising from the basic fields of the theory. This paper is organized as follows: in Sec. 2 we will give a brief review of the CPI which was presented at this conference by Ennio Gozzi. In Sec. 3 we will extend the formalism of the CPI from the point particle to a field theory of fermions. In Sec. 4 we will briefly review the analysis of the chiral anomaly in the quantum path integral approach a` la Fujikawa [4]: the breaking of the chiral symmetry arises because the functional measure is not invariant under the chiral transformations. In Sec. 5 we will implement the chiral symmetry for Fujikawa's models at the CPI level. The associated functional measure turns out to have a part identical to the quantum one and another part which involves a functional integral over the auxiliary fields which appear in the CPI. These auxiliary fields play a crucial role in compensating the anomaly coming from the quantum part of the functional measure and in restoring the chiral symmetry at the classical level. 1 D. Mauro 2 Brief review of the classical path integral In this section we will limit ourselves to review those features of the formalism which are crucial in order to understand the extension to the field theories. For further details we refer the interested reader to the original papers [1]. It is well-known from Koopman and von Neumann's work [5] that it is possible to formulate classical mechanics using states and operators defined in a suitable Hilbert space on phase space whose coordinates will be indicated with ' ≡ (q; p). The main idea of this approach is to replace the probability densities ρ(') with the states of a Hilbert space ('), whose modulus square reproduces just the probability density of finding the system in a certain point of the phase space, i.e. j (')j2 = ρ('). The evolution of these \Koopman{von Neumann waves" is given by the so called Liouville equation: @ i = L^ ; (1) @t where the Liouvillian L^ can be written in terms of the Hamiltonian H(') and of the ab 0 1 ab antisymmetric matrix ! = as follows: L^ = i @aH! @b. Because of −1 0 the particular form of the operator of evolution, which is first order in the derivatives with respect to q and p, it is easy to prove that also the probability densities ρ(') evolve with the Liouville equation (1). Now, since every theory formulated with operatorial techniques can be rewritten in the path integral language, it must be possible to reformulate also classical mechanics via path integrals. This has been done in Ref. [1] by starting from the following question: which is the probability density of going from the point 'i of the phase space at time ti to the point 'f at time tf ? In classical mechanics we have only two possibilities: this probability is a one if the point 'f lies at time tf on the classical path φcl(t; 'i) and zero otherwise. a By classical path φcl(t; 'i) we mean the path which solves the classical Hamilton's a ab equations of motion '_ = ! @bH(') with the initial condition '(ti) = 'i. This result can be written as a path integral over ' of a functional Dirac delta which gives weight one only to the classical path associated with the initial conditions 'i. 00 a a Z = h'f ; tf j'i; tii = D ' δ [' − φ (t; 'i)] : (2) Z cl The double prime on D means that the initial and the final point in the phase space are fixed. If we now use the properties of the Dirac deltas we can replace the Dirac delta on the solutions of the equations of motion with a Dirac delta on the equations of motion plus a functional determinant: a a a ab a ab δ [' − φcl(t; 'i)] = δ('_ − ! @bH) det(@tδc − ! @b@cH) : (3) We can use the Fourier representation of the Dirac delta to exponentiate the Dirac delta on the equations of motion via a functional integral over an auxiliary variable λa. Furthermore we can use the Faddeev{Popov trick to exponentiate the determi- a nant in the RHS of (3) via a couple of Grassmannian odd variables c and c¯a. The 2 Cancellation of anomalies in a path integral formulation for classical field theories final result is that the probability amplitude (2) can be rewritten as the following path integral: tf Z = D00'DλDcDc¯ exp i dt L ; (4) Z Zti e where the functional integral is extended not only over the phase space variables ' but also over all the auxiliary variables λ, c and c¯. The Lagrangian L is given by a a ab ab ed L = λa'_ + ic¯ac_ − H ; H = λa! @bH + ic¯a! @b@dHc : Let us nowedefine the commutators as Feynman did for quantum mechanics, i.e. using the following rule h[O1(t); O2(t)]i ≡ limhO1(t + )O2(t) O2(t + )O1(t)i : !0 We get that ['^a; '^b] = 0, i.e. the position q^ and the momentum p^ commute, which confirms that we are doing classical and not quantum mechanics. The only non-zero graded commutators are given by: a ^ a a a ['^ ; λb]− = iδb ; [c^ ; ^c¯b]+ = δb : The previous commutators can be realized by taking '^a and c^a as multiplicative operators and λ^a and ^c¯a as the following derivative operators: @ @ λ^ = −i ; ^c¯ = : a @'a a @ca Via this choice of operators the Hamiltonian which appears in the weight of the classical path integral (4) becomes the following operator: @ H^ = −i!ab@ H@ − i!ab@ @ Hcd : b a b d @ca The first operator is just the Liouvillian which enters the equation of evolution (1) of the \Koopman{von Neumann waves". In this sense we can say that the path integral (4) can be considered as the functional counterpart of the Koopman{von Neumann formalism. This path integral formulation of classical mechanics is very rich from the geometrical point of view and part of this richness has been explored in Ref. [6]. As now classical mechanics has been formulated using the same tools of quantum mechanics, it is easy to make a comparison between the two theories [7] and to study the relative interplay. A typical example of the interplay between classical and quantum mechanics in field theories is given by the issue of anomalies, i.e. symmetries which are present at the classical level but that are broken by the quantization procedure. For example [3] a field theory of fermions coupled with a gauge field is invariant under chiral transformations in classical mechanics but leads to a chiral anomaly at the quantum level. This talk is based just on Ref. [3] to which we refer the interested reader for further technical details. 3 D. Mauro 3 Classical path integral for fermions Since the main goal of this paper is to study the chiral symmetry in the frame- work of the CPI, the first thing that we have to do is to extend the formalism of the CPI from the point particle case, that we have briefly reviewed in Sec. 2, to the case of a field theory of fermions endowed with chiral symmetry. Let us start from the simple case of a free massless fermion theory. The Lagrangian of the system is given by: µ L = i dx ¯(x)γ @µ (x) ; (5) Z where (x) is a Grassmannian odd field and ¯(x) is defined as ¯ = yγ0. The Hamiltonian associated with the Lagrangian (5) is: y 0 l H = −i dx (x)γ γ @l (x) : Z The Euler{Lagrange equations which can be derived from (5) are: 0 l y y 0 l _ + γ γ @l = 0 ; _ + @l γ γ = 0 : (6) From (5) we have that y can be considered as the momentum canonically conju- gated to .